The numerical approach to the study of jet dynamics
is an approximation of the real physical situations. In fact, although
numerical simulations have become very sophisticated owing to
supercomputers, nevertheless,
the solution of the set of Navier-Stokes or Euler plus Maxwell equations
is still limited to relatively low Reynolds numbers (high viscosity)
R e
100 owing to the discretization process by finite difference schemes. On
the other hand, laboratory experiments indicate that for highly supersonic
jets above these values of Re, the physics does not change
appreciably from the phenomenological point of view.

HYDRODYNAMIC SIMULATIONS
Temporal analysis Early simulations of the evolution of
instabilities in cylindrical or slab symmetry were presented by
Hardee & Norman
(1988) but were rather limited in time.
Bodo et al (1994,
1995),
Basset & Woodward
(1995)
have followed the evolution of unstable modes in infinite 2.5-D cylindrical
jets and 2-D slabs. By applying periodic boundary conditions at the initial
and final cross sections of the jet at the extremes of the integration
domain, they simulated the evolution of local perturbations of
wavelengths shorter than the domain length in an infinite flow. This is
called a temporal analysis of the instability. In this way, the
instability can be followed to see whether
nonlinear saturation effects yield to the onset of a quasistationary state.
Bodo et al (1994),
following an exploration of the relevant parameters, i.e. density
contrast =
ext
/ j
and Mach number Mj, determined that jets, after a time
t ~ 30Rj / cs, reach a
quasistationary highly turbulent configuration. Heavy jets
maintain a coherent directionality; light jets appear completely mixed and
diffused. In conclusion, the persistence of jets depends principally on the
density contrast with the ambient medium. Examples of the evolution are
illustrated in Figure 5a, b.

The situation changes drastically in 3-D simulations of cylindrical jets
(Bodo et al 1998).
Mixing starts much earlier owing to the more rapid growth of small-scale
structures, and this is particularly evident in light jets, where fluting
modes are present in the linear phase with growth rates that are already
larger than those of helical modes. Light jets are asymptotically disrupted
by a strong transition to turbulence. Dense jets survive as collimated
structures, although the energy and momentum lost by entrainment is
larger and the process occurs faster, over time scales of t
~ 10Rj / cs. The final flow velocity
also is reduced, and the jet cross section
is broadened, as is consistent with strong mass entrainment, which can reach
the same load of the jet mass (see
Figure 5c).

The physical reason for the instability enhancement in
3-D geometry is the faster development of small-scale structures
independently from the initial perturbation. These can be either excited
directly, owing to the large growth rate of nonaxisymmetric 3-D modes,
or indirectly, through
the nonlinear turbulent cascade of energy from large- to small-scale eddies.
Another effect that appears to be important in this respect is the different
scaling of volumes that makes 3-D jets more expanded.

On the other hand, recent numerical results appear to
confirm a result of linear calculations, which shows that the presence of
an extended layer around the jet can stabilize the flow by suppressing
perturbations with scales smaller than the transverse dimension of the
layer. As we discuss
in the next section, the formation of layers is admissible under the form
of a cocoon produced by the bow shock of the advancing head of the jet or
to nonlinear fluid effects at the flow boundary.

Spatial analysis
In a different numerical approach, the flow is considered
as a finite window on an infinite jet with free boundary conditions at the
extremes of the integration domain, and perturbations are produced at the
injection nozzle. These perturbations are then followed in their spatial
growth while crossing the integration domain and passing through a still
undisturbed medium. In this way, nonlinear spatial effects can be analyzed
as, for instance, the interaction and merging of shocks along the jet
(Norman et al 1988,
Micono et al 1998).

The nonlinear evolution of spatial axisymmetric perturbations
in 2-D cylindrical and slab structures essentially agrees with the temporal
analysis in its initial three stages. The axisymmetric perturbation that
dominates eventually is the first reflected mode that actually has the
fastest
spatial growth rate. A tendency is observed of coalescence of successive
shocks into an almost transverse single strong shock that extends into the
external medium through entrainment and momentum dissipation.

Antisymmetric perturbations in cylindrical
jets or nonaxisymmetric perturbations in slabs instead create piston-like
protrusions (spurs) into the external medium that travel along the
integration grid. They can never reach the quasistationary stage because
the spur amplitude becomes very large while travelling along the jet and
in fact disrupts the ordered flow
(Norman et al 1988,
Micono et al 1998).
In addition, longitudinal filamentary structures develop that can wrap
around the jet if rotation is included in the calculations
(Hardee & Stone
1997).

Cooling jets
A crucial question is whether radiative losses can
affect the global evolution of the instability. In most 2-D cases, even when
counteracted by heating, they slow down the growth of instability
(Rossi et al 1997,
Stone et al 1997,
Micono et al 1998).
In cylindrical geometry, thermal losses (a case that best applies to stellar
jets) are very efficient in suppressing mixing of the jet matter with the
external medium, and subsequently matter entrainment. Shocks remain well
separated and maintain the characteristic zigzag pattern. However, mixing
is present in dense jets. For a slab, mixing and shock coalescence develop
on short time scales, and the growth of the instability may in fact be
faster. In the case of synchrotron losses, which are more appropriate to
extragalactic jets, in addition to Kelvin-Helmholtz-type instabilities,
filamentation modes related to thermal-type instabilities are most
important and modulate the jets longitudinally
(Rossi et al 1993).
In fully 3-D geometries that are characterized by faster growth of the
instability, radiation losses appear to be too slow and unable to stop
the disruption of jets.

MHD SIMULATIONS
From a historical point of view, we recall that
a 2-D MHD particle code was used by
Tajima & Leboeuf
(1980)
to study Kelvin-Helmholtz instabilities of a single shear layer parallel
to a uniform magnetic field but did not reach long time scales of evolution.
The numerical analysis of nonlinear MHD instabilities is still limited to
rather simple configurations. Most experiments performed so far have used
the standard finite difference scheme with finite cells and,
correspondingly, a rather large numerical viscosity.
Shibata & Uchida
(1986)
used an evolved Lax-Wendroff scheme. Stone, Norman, and collaborators
(Stone & Norman
1992,
1994)
developed a 2-D MHD code named ZEUS that is partly based on a higher-order
upwind integration scheme. An extension of this last code to a 3-D case is
now available but has low resolution for studying the formation of vortices
and turbulence. As a consequence, these simulations tend to smooth out
strong instabilities and discontinuities.

Recently,
Zachary et al (1994)
have succeeded in producing a 2-D MHD code with parabolic upwind integration
along the characteristics across discontinuities. This MHD Godunov code has
been tested on the standard problem of the Kelvin-Helmholtz instability of
a shear layer in the case cA <
cs (large magnetic fields suppress the instability)
(Malagoli et al 1996);
the formation of cats' eyes has been followed, and the subsequent series
of reconnection events asymptotically yields a stationary turbulent thick
layer (Figure 6).
A relatively small magnetic field, well below equipartition, helps the shear
layer reach a stationary state in which the two fluids in relative motion
are separated by a turbulent boundary sheath that eliminates direct
interaction.
Thus, reconnection seems to be the crucial physical process in governing
the magnetic instability evolution. Similar results have been obtained by
Frank et al (1996),
Min (1997) using an
FCT code.

SHOCKS We conclude discussing in more detail
the evolution of "internal shocks" already predicted in linear studies but
now properly followed by numerical methods. These shocks arise in the form
of conical structures inside a cylindrical jet with typical opening angles
of ~ 1 / Mj. The repetitive pattern of oblique shocks
is a typical feature of the nonlinear evolution of jets. All these
shocks travel with the flow at a velocity
slightly below the jet velocity and may be related, as shown in
Section 6,
to emission morphologies. The intersection points of shocks correspond to
high pressure regions with strong emission. Actually, as shown by
Hardee & Norman
(1989),
the merging of shocks gives rise to phase effects where the intersection
points can move at a velocity higher than the jet's. This is clearly
relevant
to the interpretation of superluminal motions in relativistic jets. In later
stages, owing to mass entrainment and momentum diffusion, shocks extend at
large distances into the external medium perpendicular to the flow and
become substantial transverse structures
(Bodo et al 1994).